Byeong-Joo Lee

Byeong-Joo Lee “Numerical Treatment of Moving Interface in Diffusional Reactions,” Byeong-Joo Lee and Kyu Hwan Oh, Z. Metallkunde 87, (1996). “Numerical Procedure for Simulation of Multicomponent and Multi-Layered Phase Diffusion,” Byeong-Joo Lee, Metals and Materials 5, 1-15 (1999). “Numerical Simulation of Diffusional Reactions between Multiphase Alloys with Different Matrix Phases,” Byeong-Joo Lee, Scripta Materialia 40, (1999). “Prediction of the Amount of Retained delta-ferrite and Microsegregation in an Austenitic Stainless Steel,” Byeong-Joo Lee, Z. Metallkunde 90, (1999). “Evaluation of Off-Diagonal Diffusion Coefficient from Phase Diagram Information,” Byeong-Joo Lee, J. Phase Equilibria 22, (2001). “Thermo-Calc & DICTRA, computational tools for materials science,” J.-O. Andersson, Thomas Helander, Lars Höglund, Pingfang Shi and Bo Sundman, CALPHAD 26, (2002) References

Byeong-Joo Lee Diffusional Reactions – binary & multicomponent systems

Byeong-Joo Lee Multicomponent Diffusion Fe-3.8Si-C Fe-C Darken’s uphill diffusion Diffusion between multiphase layers A. Engström, Scand. J. Metall. 24, 12 (1995). B.-J. Lee, J. Phase Equilibria 22, 241 (2001).

Byeong-Joo Lee Content 1. Introduction ․ Definition ․ Diffusion Mechanism: Vacancy Mechanism, Interstitial Mechanism 2. Diffusional Flux and Application of Fick's law ․ Fick's first law in two component system ․ Fick's second law Application - Steady State Solution 3. Non-Steady State Diffusion ․ Thin Film Source (Thin Layer) ․ Semi-Infinite Source (Diffusion Couple) ․ Laplace/Fourier Transformation ․ Error function ․ Homogenization/Solute penetration ․ Trigonometric-Series Solutions ․ Determination of diffusion coefficient (Grube, Boltzman-Matano method) ․ Other Examples ․ Diffusion along high diffusion paths 4. Diffusion Coefficients ․ Reference Frame of Diffusion ⇒ Darken's Equation ․ Intrinsic, Inter, Self, Trace, Impurity Trace Diffusion Coefficient ․ Reference : Smithells Metals Reference Book, Chap. 13., Reed-Hil 5. Modelling of Multicomponent Diffusion ․ Darken's experiments : Fe-Si-C ․ Mathematical Formalism for Multicomponent Diffusion Coefficient

Byeong-Joo Lee Definition Homogenization phenomena by non-convective mass transport due to chemical potential or electrochemical potential difference in a multicomponent single phase

Byeong-Joo Lee General Phenomenological Equation

Byeong-Joo Lee Fick’s 1 st law

Byeong-Joo Lee Fick’s 2 nd law

Byeong-Joo Lee As a thermally activated process for interstitial diffusion More about Diffusion Coefficient – Thermal Activation How about for substitutional diffusion?

Byeong-Joo Lee Steady State Solution of Diffusion

Byeong-Joo Lee Non-Steady State Solution of Diffusion

Byeong-Joo Lee Non-Steady State Solution of Diffusion - Superposition Principle

Byeong-Joo Lee Non-Steady State Solution of Diffusion - Superposition Principle

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Application of Superposition Principle

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Leak Test & Error Function

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Semi-Infinite Source

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Semi-Infinite Source

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Semi-Infinite Source

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Semi-Infinite Source

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Semi-Infinite Source

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Semi-Infinite Source

Byeong-Joo Lee Determination of Diffusivity – Grube method

Byeong-Joo Lee Determination of Diffusivity – Boltzmann-Matano

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Separation of Variable

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Separation of Variable

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Separation of Variable

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Separation of Variable

Byeong-Joo Lee Non-Steady State Solution of Diffusion – Separation of Variable

Byeong-Joo Lee Diffusion along High Diffusion Path – Grain Boundary Diffusion Model

Byeong-Joo Lee Diffusion Simulation – Finite Difference Method

Byeong-Joo Lee Diffusion Simulation – Finite Difference Method

Byeong-Joo Lee Diffusion Simulation – Finite Difference Method implicit integer (i-n) implicit double precision (a-h,o-z) dimension U(1000), UF(1000) c write(*,'(a)',advance='NO') ' Length of Simulation (micro-m) ? ' read(*,*) XL write(*,'(a)',advance='NO') ' Initial Composition (U-fraction) ? ' read(*,*) Uini write(*,'(a)',advance='NO') ' Boundary (Left-end) Composition ? ' read(*,*) U0 write(*,'(a)',advance='NO') ' Diffusion Coefficient (cm^2/sec) ? ' read(*,*) D write(*,'(a)',advance='NO') ' Reaction Time (sec) ? ' read(*,*) Tend write(*,'(a)',advance='NO') ' number of grid ? ' read(*,*) n c D = 1.d+08 * D dx = XL / dble(n-1) dt = 0.25d0 * dx * dx / D dtdx = D * dt / dx / dx c xiter = Tend / dt nprnt = idint(xiter/10.d0) c c initial condition c U = Uini UF = Uini time = 0.d0 iter = 0 open(unit=1,file='result.exp',status='unknown') write(1,'(a,f12.6)') '$ time = ', time write(1,'(f6.2,f12.6,a)') 0.d0, uf(1), ' M' do i = 2, n write(1,'(f6.2,f12.6)') dble(i-1)*dx, uf(i) enddo c c Boundary condition U(1) = U0 UF(1) = U0 U(n+1) = U(n-1) c 1 iter = iter + 1 time = time + dt do i = 2, n uf(i) = u(i) + dtdx * ( u(i+1) - 2.d0*u(i) + u(i-1) ) enddo uf(n+1) = uf(n-1) c u = uf c if(mod(iter,nprnt).eq. 0) then write(1,'(a,f12.6)') '$ time = ', time write(1,'(f6.2,f12.6,a)') 0.d0, uf(1), ' M' do i = 2, n write(1,'(f6.2,f12.6)') dble(i-1)*dx, uf(i) enddo endif c if(time.lt.tend) goto 1 stop end

Byeong-Joo Lee Diffusion Simulation – Finite Difference Method

Byeong-Joo Lee Diffusion Coefficient – Inter Diffusion

Byeong-Joo Lee Diffusion Coefficient – Inter Diffusion

Byeong-Joo Lee Diffusion Coefficient – Self/Tracer Diffusion

Byeong-Joo Lee Diffusion Coefficient – Intrinsic Diffusion Coefficient

Byeong-Joo Lee Diffusion Coefficient – Inter Diffusion Coefficient

Byeong-Joo Lee Inter-diffusion Coefficient in a binary alloy – linked to intrinsic diffusion by the Darken’s relation Intrinsic diffusion Coefficient – composed of mobility term (Tracer Diffusion) and thermodynamic factor Tracer diffusion Coefficient – as a function of composition & temp. : tracer impurity diffusion coefficient : self-diffusion of A in the given structure Diffusion Coefficient – Modeling

Byeong-Joo Lee Diffusion Coefficient – Modeling  assuming composition independent D o Linear composition dependence of Q B in a composition range N 1 ~ N 2  Tracer diffusion Coefficient at an intermediate composition is a geometrical mean of those at both ends – from experiments  the same for the D o term  Both Q o & Q are modeled as a linear function of composition

Byeong-Joo Lee Moving Boundary Problem – Basic Equation

Byeong-Joo Lee Binary Diffusion

Byeong-Joo Lee Modeling of Multi-Component Diffusion - Basic Assumption for k  S (substitutional) for k  S

Byeong-Joo Lee Modeling of Multi-Component Diffusion - Reference Frame

Byeong-Joo Lee Mathematical Formalism of Multi-Component Diffusion Coefficient

Byeong-Joo Lee Mathematical Formalism of Multi-Component Diffusion Coefficient

Byeong-Joo Lee Mathematical Formalism - Application to Binary and Ternary Solutions

Byeong-Joo Lee Mathematical Formalism - Application to Binary and Ternary Solutions

Byeong-Joo Lee Smithells Metals Reference Book, 1992 Mathematical Formalism - Application to Binary and Ternary Solutions

Byeong-Joo Lee John Ågren, Scripta Metallurgica 20, (1986). Mathematical Formalism - Application to Binary and Ternary Solutions

Byeong-Joo Lee Mathematical Formalism - Application to Binary and Ternary Solutions

Byeong-Joo Lee Multi-Component Diffusion Simulation – for C in Fe-C-M ternary system

Byeong-Joo Lee Multi-Component Diffusion Simulation – Darken’s uphill diffusion Fe-3.8Si-C and Fe-C

Byeong-Joo Lee Multi-Component Diffusion Simulation – Darken’s uphill diffusion

Byeong-Joo Lee Multi-Component Diffusion Simulation – Darken’s uphill diffusion Fe-3.8Si-C and Fe-6.45Mn-C

Byeong-Joo Lee Multi-Component Diffusion Simulation – Darken’s uphill diffusion

Byeong-Joo Lee Multi-Component Diffusion Simulation – FDM approach for Fe-Si-C

Byeong-Joo Lee Moving Boundary Problem – Basic Equation

Byeong-Joo Lee Binary Diffusion

Byeong-Joo Lee Application to Interfacial Reactions – Ti/Al 2 O 3 Reaction

Byeong-Joo Lee Application to Interfacial Reactions – Ti/Al 2 O 3 Reaction

Byeong-Joo Lee Multi-Component Diffusion Simulation – Case Study : Fe-Cr-Ni

Byeong-Joo Lee Multi-Component Diffusion Simulation – Case Study : Fe-Cr-Ni

Byeong-Joo Lee Multi-Component Diffusion Simulation – between Multi-Phase Layers

Byeong-Joo Lee Multi-Component Diffusion Simulation – between Multi-Phase Layers

Byeong-Joo Lee Multi-Component Diffusion Simulation – between Multi-Phase Layers